Chapter 14 Geophysical Fluid Dynamics 1. Introduction ........................ 603 11. Gravity Waves with Rotation ......... 636 2. Vertical Variation of Density in Particle Orbit ....................... 638 Atmosphere and Ocean .............. 605 Inertial Motion ...................... 639 3. Equations of Motion ................. 607 12. Kelvin Wave ......................... 639 Formulation of the Frictional 13. Potential Vorticity Conservation in Term............................. 608 Shallow-Water Theory............... 644 4. Approximate Equations for a Thin 14. Internal Waves ...................... 647 Layer on a Rotating Sphere.......... 610 WKB Solution ....................... 650 f -Plane Model ...................... 612 Particle Orbit ....................... 652 β-Plane Model ...................... 612 Discussion of the Dispersion 5. Geostrophic Flow.................... 613 Relation. ......................... 654 Thermal Wind....................... 614 Lee Wave . 656 Taylor–Proudman Theorem .......... 615 15. Rossby Wave ........................ 657 6. Ekman Layer at a Free Surface ...... 617 Quasi-geostrophic Vorticity Explanation in Terms of Equation. ........................ 658 Vortex Tilting .................... 622 Dispersion Relation .................. 660 7. Ekman Layer on a Rigid Surface ..... 622 16. Barotropic Instability................ 663 8. Shallow-Water Equations ............ 625 17. Baroclinic Instability ................ 665 9. Normal Modes in a Continuously Perturbation Vorticity Equation ...... 666 Stratified Layer ..................... 628 Wave Solution ....................... 668 Boundary Conditions on ψn .......... 630 Boundary Conditions ................ 669 Solution of Vertical Modes for Instability Criterion.................. 669 Uniform N ....................... 631 Energetics........................... 671 10. High- and Low-Frequency Regimes in 18. Geostrophic Turbulence ............. 673 Shallow-Water Equations............ 634 Exercises ............................ 676 Literature Cited ..................... 677 1. Introduction The subject of geophysical fluid dynamics deals with the dynamics of the atmosphere and the ocean. It has recently become an important branch of fluid dynamics due to our increasing interest in the environment. The field has been largely developed by meteorologists and oceanographers, but non-specialists have also been interested in the subject. Taylor was not a geophysical fluid dynamicist, but he held the position of 603 604 Geophysical Fluid Dynamics a meteorologist for some time, and through this involvement he developed a special interest in the problems of turbulence and instability. Although Prandtl was mainly interested in the engineering aspects of fluid mechanics, his well-known textbook (Prandtl, 1952) contains several sections dealing with meteorological aspects of fluid mechanics. Notwithstanding the pressure for specialization that we all experience these days, it is worthwhile to learn something of this fascinating field even if one’s primary interest is in another area of fluid mechanics. The importance of the study of atmospheric dynamics can hardly be overem- phasized. We live within the atmosphere and are almost helplessly affected by the weather and its rather chaotic behavior. The motion of the atmosphere is intimately connected with that of the ocean, with which it exchanges fluxes of momentum, heat and moisture, and this makes the dynamics of the ocean as important as that of the atmosphere. The study of ocean currents is also important in its own right because of its relevance to navigation, fisheries, and pollution disposal. The two features that distinguish geophysical fluid dynamics from other areas of fluid dynamics are the rotation of the earth and the vertical density stratification of the medium. We shall see that these two effects dominate the dynamics to such an extent that entirely new classes of phenomena arise, which have no counterpart in the laboratory scale flows we have studied in the preceding chapters. (For example, we shall see that the dominant mode of flow in the atmosphere and the ocean is along the lines of constant pressure, not from high to low pressures.) The motion of the atmosphere and the ocean is naturally studied in a coordinate frame rotating with the earth. This gives rise to the Coriolis force, which is discussed in Chapter 4. The density stratification gives rise to buoyancy force, which is introduced in Chapter 4 (Conservation Laws) and discussed in further detail in Chapter 7 (Gravity Waves). In addition, important relevant material is discussed in Chapter 5 (Vorticity), Chapter 10 (Boundary Layer), Chapter 12 (Instability), and Chapter 13 (Turbulence). The reader should be familiar with these before proceeding further with the present chapter. Because Coriolis forces and stratification effects play dominating roles in both the atmosphere and the ocean, there is a great deal of similarity between the dynam- ics of these two media; this makes it possible to study them together. There are also significant differences, however. For example the effects of lateral boundaries, due to the presence of continents, are important in the ocean but not in the atmosphere. The intense currents (like the Gulf Stream and the Kuroshio) along the western boundaries of the ocean have no atmospheric analog. On the other hand phenomena like cloud formation and latent heat release due to moisture condensation are typically atmo- spheric phenomena. Processes are generally slower in the ocean, in which a typical horizontal velocity is 0.1 m/s, although velocities of the order of 1–2 m/s are found within the intense western boundary currents. In contrast, typical velocities in the atmosphere are 10–20 m/s. The nomenclature can also be different in the two fields. Meteorologists refer to a flow directed to the west as an “easterly wind” (i.e., from the east), while oceanographers refer to such a flow as a “westward current.” Atmospheric scientists refer to vertical positions by “heights” measured upward from the earth’s surface, while oceanographers refer to “depths” measured downward from the sea surface. However, we shall always take the vertical coordinate z to be upward, so no confusion should arise. 2. Vertical Variation of Density in Atmosphere and Ocean 605 We shall see that rotational effects caused by the presence of the Coriolis force have opposite signs in the two hemispheres. Note that all figures and descriptions given here are valid for the northern hemisphere. In some cases the sense of the rotational effect for the southern hemisphere has been explicitly mentioned. When the sense of the rotational effect is left unspecified for the southern hemisphere, it has to be assumed as opposite to that in the northern hemisphere. 2. Vertical Variation of Density in Atmosphere and Ocean An important variable in the study of geophysical fluid dynamics is the density strat- ification. In equation (1.38) we saw that the static stability of a fluid medium is determined by the sign of the potential density gradient dρpot dρ gρ , (14.1) dz = dz + c2 where c is the speed of sound. A medium is statically stable if the potential density decreases with height. The first term on the right-hand side corresponds to the in situ density change due to all sources such as pressure, temperature, and concentration of a constituent such as the salinity in the sea or the water vapor in the atmosphere. The second term on the right-hand side is the density gradient due to the pressure decrease with height in an adiabatic environment and is called the adiabatic density gradient. The corresponding temperature gradient is called the adiabatic temperature gradient. For incompressible fluids c and the adiabatic density gradient is zero. As shown in Chapter 1, Section=∞ 10, the temperature of a dry adiabatic atmosphere decreases upward at the rate of 10 ◦C/km; that of a moist atmosphere decreases ≈ 2 at the rate of 5–6 ◦C/km. In the ocean, the adiabatic density gradient is gρ/c 3 ≈4 4 10− kg/m , taking a typical sonic speed of c 1520 m/s. The potential density ∼ × = 3 4 in the ocean increases with depth at a much smaller rate of 0.6 10− kg/m , so that the two terms on the right-hand side of equation (14.1) are× nearly in balance. It follows that most of the in situ density increase with depth in the ocean is due to the compressibility effects and not to changes in temperature or salinity. As potential density is the variable that determines the static stability, oceanographers take into account the compressibility effects by referring all their density measurements to the sea level pressure. Unless specified otherwise, throughout the present chapter potential density will simply be referred to as “density,” omitting the qualifier “potential.” The mean vertical distribution of the in situ temperature in the lower 50 km of the atmosphere is shown in Figure 14.1. The lowest 10 km is called the troposphere, in which the temperature decreases with height at the rate of 6.5 ◦C/km. This is close to the moist adiabatic lapse rate, which means that the troposphere is close to being neutrally stable. The neutral stability is expected because turbulent mixing due to frictional and convective effects in the lower atmosphere keeps it well-stirred and therefore close to the neutral stratification. Practically all the clouds, weather changes, and water vapor of the atmosphere are found in the troposphere. The layer is capped by the tropopause, at an average height of 10 km, above which the temperature